The sharp constant in Jackson-type inequality for approximation by linear positive operators

O. L. Vinogradov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 26, Tome 255 (1998), pp. 36-53

Résumé

In what follows, $C$ is the space of $2\pi$-periodic continuous real-valued functions with uniform norm, $\omega(f,h)=\sup_{|t|\le{h},x\in\mathbb R}|f(x+t)-f(x)|$ is the first modulus of continuity of function $f\in C$ with step $h$, $H_n$ is the set of trigonometric polynomials of order not greater than $n$, ${\mathscr L}_n^+$ is the set of linear positive operators $U_n:C\to H_n$ (i.e. such that $U_n(f)\ge0$ for every $f\ge0$), $L_2[0,1]$ is the space of square integrable on $[0,1]$ functions, $$ \lambda_n(\gamma)=\inf_{U_n\in{\mathscr L}_n^+}\sup_{f\in C}\frac{\|f-U_n(f)\|}{\omega(f,\frac{\gamma\pi}{n+1}}, \qquad \lambda(\gamma)=\sup_{n\in\mathbb Z_+}\lambda_n(\gamma). $$ It is proved that $\lambda_n(\gamma)$ coincides with the smallest eigenvalue of some matrix of order $n+1$. The principal result of the paper is the following: for every $\gamma>0$ $\lambda(\gamma)$ doesn't outnumber and for $\gamma\in(0,1]$ is equal to the minimum of square functional $$ (B_{\gamma}\varphi,\varphi)=\frac1\pi\int\limits_0^{\infty}\biggl(1+\biggl[\frac{t}{\gamma\pi}\biggr]\biggr)\Biggl|\int\limits_0^1\varphi(x)e^{itx}\,dx\Biggr|^2dt $$ on the unit sphere of $L_2[0,1]$. Then it is calculated that $\lambda(1)=1.312\ldots$